Jim Burns wrote:> On 8/16/2013 4:54 AM, Peter Percival wrote:>> Can addition be defined in terms of multiplication? I.e.,>> is there a formula in the language of arithmetic>>>> x + y = z <-> ...>>>> such that in '...' any of the symbols of arithmetic>> except + may occur? Or, alternatively, is there a>> formula in the language of arithmetic>>>> x + y = ...>>>> with the same requirement?>>>> The symbols of arithmetic (for the purpose of this question) are either>>>> individual variables, (classical) logical constants including =,>> S, +, *, and punctuation marks;>>>> or the above with < as an additional binary predicate symbol.>> How about>> x + y = z <-> 2^x * 2^y = 2^z>> where 2^x is just an abbreviation for the function 2pwr: N -> N,> defined by> 2pwr(0) = 1>> 2pwr( Sx ) = 2 * 2pwr( x )

That goes beyond what I defined as the language of arithmetic.

-- Sorrow in all lands, and grievous omens.Great anger in the dragon of the hills,And silent now the earth's green oraclesThat will not speak again of innocence. David Sutton -- Geomancies